NCERT Class 9 Ganita Manjari Chapter 06 Measuring Space Perimeter and Area PDF Download

Chapter 6: Measuring Space: Perimeter and Area

[Figure 6.1: Athletes at the start of 4 × 100 m relay race, See in your textbook]

In Fig. 6.1, you see athletes assembled at the start of a 4 × 100 m relay race. The tracks are laid out, and the athletes are all set to go racing down the tracks. Do you notice that the athletes are not at the same starting line?

Those in the outer lanes seem to be starting ahead of those in the inner lanes while the finish line is the same for all of them. What could be the reason for this? The distance between the starting points of adjacent lanes is called the 'stagger'. Notice that the stagger continues all the way to the outermost lane. Do you think the stagger gives anyone (those in the outer lanes or in the inner lanes) an unfair advantage? Why or why not? On what basis can the organisers work out the length of the stagger between lanes?

Think and Reflect

In my school, the playground is too small to have a 400 m track, so the school constructed a 200 m track instead. Does this mean that we need a smaller stagger for the race tracks in my school (i.e., smaller than the stagger used in the Olympics), for the same 4 × 100 m relay race?

To answer the question about the lane staggers required on a 4 × 100 m athletics track, we need to know how to find the length around a circle.

6.1 Perimeter of a Shape

Given any shape, its perimeter is the total length around its border. Imagine a tiny insect going for a walk around its border, never turning around, till it returns to its starting point. The perimeter of the shape is the total distance it travels.

So, a square with side \( a \) units has perimeter \( 4a \) units. An equilateral triangle with side \( a \) units has perimeter \( 3a \) units.

[Figure 6.2A: Square with side a units, See in your textbook]

[Figure 6.2B: Equilateral triangle with side a units, See in your textbook]

The perimeter of a rectangle with length \( a \) units and width \( b \) units is \( 2(a + b) \) units. Note that the formula for the perimeter of a square is a 'special case' of the formula for the perimeter of a rectangle with \( a = b \).

[Figure 6.3: Circle with radius r units, See in your textbook]

Here we see a circle with radius \( r \) units. What is its perimeter? How do we find out?

Think and Reflect

What is the connection between this question and the one about the 400 m athletics track?

To answer the question about the perimeter of the circle, we must go step by step.

What happens to the perimeter of a square if we double its side? It doubles too. The ratio of perimeter to the side is 4:1; this is so for all squares. As the side of the square gets larger (or smaller), the ratio of perimeter to side stays fixed at 4:1.

[Figure 6.4: Ratio of perimeter to side is 4:1, See in your textbook]

Teacher's Note

When you work with perimeters, always check that your answer makes sense in relation to the side length. If the side is in centimetres, the perimeter will also be in centimetres. Remember that a square with side 5 cm has perimeter \( 4 \times 5 = 20 \) cm, not \( 5 \times 5 = 25 \) cm - the second is the area, which is different.

For equilateral triangles, the ratio of perimeter to the side is 3:1. As the side of the equilateral triangle gets larger (or smaller), the ratio of perimeter to side stays fixed at 3:1.

[Figure 6.5: Three circles showing C/D ratio, See in your textbook]

What about a circle? What is its perimeter (usually called the circumference) in terms of its diameter?

Is the ratio of circumference (C) to diameter (D) the same for circles of all sizes (Fig. 6.5)? What do you think?

6.2 Perimeter of a Circle - The C/D Ratio

In ancient days people realised that the ratio of the circumference to the diameter of the circle does not change if we change the size of the circle.

Let's call this ratio the 'C/D ratio' of the circle.

What is the value of the C/D ratio?

How would you estimate this ratio?

Home Measurement

You can do a simple measurement at home to estimate the C/D ratio. Take a cotton reel with thin thread around it. Measure the diameter \( D \) of the reel as accurately as possible. Unwrap and then tightly wrap the thread around the reel 20 times. Unwrap it again; measure its length \( L \), and calculate \( \frac{L}{20D} \). This is the ratio we want. For accuracy, the thread should be very thin. Please do the experiment! Do you get a ratio between 3 and 4? Between 3.1 and 3.2?

It is also possible to estimate the C/D ratio using pure geometry, i.e., without any measurements at all! Can you imagine how?

C/D's Adventurous Journey: From Ancient Approximations to the Exact Formula of Madhava

Mathematicians have been fascinated by circles, and the C/D ratio, since ancient times and across geographical regions. This constant,

Teacher's Note

The C/D ratio is what we call pi (\( \pi \)). When you see the circumference formula later, it will show you that C = \( \pi \) × D. Do not try to memorise the value of pi; instead, remember that it lies between 3 and 3.2, which helps you check if your answers are reasonable when working with circles.

Key Points

• The perimeter is the total distance around the border of any shape. For squares and rectangles, you add up all the side lengths.
• The ratio of perimeter to side length is fixed for shapes of the same type: 4:1 for squares, 3:1 for equilateral triangles, no matter how big or small.
• For circles, the ratio of circumference to diameter (called the C/D ratio) is also fixed at approximately 3.14, and it does not change regardless of the circle's size.

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